Using lemma REF , we obtain \(|\langle {\Omega }|{\widetilde{W}}\rangle |^2 \ge \cos ^2(\alpha _1) + \cos ^2(\alpha _2) - 1\) . We hence write \(|\langle {\Omega }|{\widetilde{W}}\rangle |^2 = \cos ^2(\beta )\)  for some \(\beta \le \alpha _1 + \alpha _2\) . In order to conclude, we define \(Angle(|\psi \rangle ,|\phi \rangle ) = \arccos (|\langle {\psi }|{\phi }\rangle |)\) . The angle function is a distance measure [1]}. We will use several times the trigonometric inequality \(\cos (\rho + \theta ) \ge \cos ^2(\rho ) + \cos ^2(\theta ) - 1\)  for any \(\rho ,\theta \)  with \(\rho + \theta \le \pi /2\)  and we can hence write
\(\left\Vert P_3 |\widetilde{W}\rangle \right\Vert ^2 & \ge |\langle {\phi _3}|{\widetilde{W}}\rangle |^2 \\& = \cos ^2\left(Angle\left(|\phi _3\rangle ,|\widetilde{W}\rangle \right)\right) \\& \ge \cos ^2\left(Angle(|\phi _3\rangle ,|\Omega \rangle ) + Angle(|\Omega \rangle ,|\widetilde{W}\rangle )\right) \\& \ge \cos ^2(\alpha _3 + \beta ) \\& \ge \left(\cos ^2(\alpha _3) + \cos ^2(\beta ) - 1\right)^2 \\& = \left(\cos ^2(\alpha _1) + \cos ^2(\alpha _2) + \cos ^2(\alpha _3) - 2\right)^2 = \varepsilon ^2\) 
